Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^{p}$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case $X=C$, we get a new approach to the $L^p$ theory of square roots of elliptic operators, as well as an $L^{p}$ version of Carleson's inequality.
We solve the Kato square root problem for general elliptic operators and systems with measurable and complex coefficients on any domain of the Euclidean space. The operators are subject to Dirichlet boundary conditions. We also allow perturbations by general complex potentials.
We develop new approach for studying the abstract Cauchy problem $\dot{x}=Ax$, $x(0)=x_0\in D(A)$ for linear operators $A$ defined on a locally convex space $X$. This approach was firstly introduced in the paper "Chernoff and Trotter type product formulas" to study the problem for Banach spaces. In this paper we not only generalize the results of the previous paper to more general topological spaces but also get new results for Banach spaces. In particular, we prove the "local" extension of Chernoff-Trotter-Kato type theorems. Applying this result, we prove Chernoff, Lie-Trotter and Trotter-Ka…
We prove that the algebraic sum of unbounded normal operators satisfies the square root problem of Kato under appropriate hypotheses. As application, we consider perturbed Schrodinger operators.
This paper is concerned with the square root problem of Kato for the "sum" of linear operators in a Hilbert space ${\mathbb H}$. Under suitable assumptions, we show that if $A$ and $B$ are respectively m-scetroial linear operators satisfying the square root problem of Kato. Then the same conclusion still holds for their "sum". As application, we consider perturbed Schrödinger operators.
In the infinite-dimensional separable complex Hilbert space we construct new abstract examples of unbounded maximal accretive and maximal sectorial operators $B$ for which ${\rm dom\,}B^{\frac{1}{2}}\ne{\rm dom\,}B^{*{\frac{1}{2}}}$. New criterions for the equality are established.
In the Trotter-Kato approximation theorem for C_0-semigroups on Banach spaces, we replace the strong by the weak operator topology and discuss the validity of the relevant implications.
We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent $\frac{1}{2}$ its fractional power domains still coincide with suitable Sobolev spaces of optimal regularity. In combination with a reduction theorem recently obtained by the authors, this solves the Kato Square Root Problem for elliptic second order operators and systems in divergence form under the same geometric assumptions.
On a domain $Ω\subseteq \mathbb{R}^d$ we consider second order elliptic systems in divergence form with bounded complex coefficients, realized via a sesquilinear form with domain $V \subseteq H^1(Ω)$. Under very mild assumptions on $Ω$ and $V$ we show that the Kato Square Root Problem for such systems can be reduced to a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends an earlier result of McIntosh to non-smooth coefficients.
We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet boundary part is Ahlfors--David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product we establish new regulari…
In this paper the general spectral properties of linear operators in Banach spaces are studied. We find sufficient conditions on structure of Banach spaces and resolvent properties that guarantee completeness of roots elements of Schatten class operators. This approach generalizes the well known result for operators in Hilbert spaces. In application, the boundary value problems for the abstract equation of second order with variable coefficients are studied. The principal part of the appropriate differential operator is not self-adjoint. The discreetness of spectrum and completeness of root el…
The Kato square root problem for divergence form elliptic operators with potential $V : \mathbb{R}^{n} \rightarrow \mathbb{C}$ is the equivalence statement $\left\Vert (L + V)^{\frac{1}{2}} u\right\Vert_{2} \simeq \left\Vert \nabla u \right\Vert_{2} + \left\Vert V^{\frac{1}{2}} u \right\Vert_{2}$, where $L + V := - \mathrm{div} A \nabla + V$ and the perturbation $A$ is an $L^{\infty}$ complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying $\left\Vert |V|^{\fracα{2}} u\right\Vert_{…
Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\geq 2$. In this article, the authors establish some weighted $L^p$ estimates of Kato square roots associated to the degenerate elliptic operators $L_w$. More precisely, the authors prove that, for $w\in A_{p}(\mathbb{R}^n)$, $p\in(\frac{2n}{n+1},\,2]$ and any $f\in C^\infty_c(\mathbb{R}^n)$, $\|L_w^{1/2}(f)\|_{L^p(w,\,\mathbb{R}^n)}\sim \|\nabla f\|_{L^p(w,\,\mathbb{R}^n)}$, where $C_c^\infty(\mathbb{R}^n)$ denotes the set…
We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a …
Let ${\bf R}$ denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator $T$ on a Banach space $X$ we prove that $T=T_M\oplus T_N$ with $T_M \in {\bf R}$ and $T_N$ quasinilpotent (nilpotent) if and only if $T$ admits a generalized Kato decomposition ($T$ is of Kato type) and $0$ is not an interior point of the corresponding spectrum $σ_{\bf R}(T)=\{λ\in \mathbb{C}: T-λ\notin {\bf R}\}$. In addition, we show that every non…
We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use $\ell_Ψ$-Hilbertian and $\infty$-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition in a Hilbert space and introduce the class of spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and $M$-cotype of a Banach space and study the properties of unconditional Schaude…
Let L(t) = --div (A(x, t)$\nabla$ x) for t $\in$ (0, $τ$) be a uniformly elliptic operator with boundary conditions on a domain $Ω$ of R d and $\partial$ = $\partial$ $\partial$t. Define the parabolic operator L = $\partial$ + L on L 2 (0, $τ$, L 2 ($Ω$)) by (Lu)(t) := $\partial$u(t) $\partial$t + L(t)u(t). We assume a very little of regularity for the boundary of $Ω$ and assume that the coefficients A(x, t) are measurable in x and piecewise C $α$ in t for some $α$ > 1 2. We prove the Kato square root property for $\sqrt$ L and the estimate $\sqrt$ L u L 2 (0,$τ$,L 2 ($Ω$)) $\approx$ $\nabla$ …
There are discussed some modifications of n-th von Neumann-Jordan constant considered by M. Kato, Y. Takahashi and K. Hashimoto. We study, among others, upper, lower, upper modified and lower modified n-th von Neumann-Jordan constant and relationships between them. There are characterized uniformly non-l_{n}^{1} Banach spaces in terms of the upper modified n-th von Neumann-Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces L^{p} and l^{p} (1\leq p\leq oo).
We shall say that a bounded linear operator $T$ acting on a Banach space $X$ admits a generalized Kato-Riesz decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and $T_N$ is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For $T$ is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator $S $ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is Riesz. We investigate generalized Drazin-Riesz invertible operators and also, characteri…
We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.
Beurling--Kato theory, Hardy--Sobolev calculus and Ritt operators
Aix-Marseille University（艾克斯-马赛大学）；CNRS, I2M Marseille（法国国家科学研究中心，I2M马赛）；Nicolaus Copernicus University（尼古拉·哥白尼大学）；Institute of Mathematics Polish Academy of Sciences（波兰科学院数学研究所）
We show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We show for example that the so-called Newton, binomial, Visser, and Halley iterative methods converge to the root in Banach and operator algebras under various mild hypotheses. We also show that the `sign' and `geometric mean' of matrices generalize to Banach and operator algebras, and we investigate their properties. We also establish some other facts about roots in this setting.
A bounded linear operator $T$ on a Banach space $X$ is said to be generalized Drazin-meromorphic invertible if there exists a bounded linear operator $S$ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is meromorphic. We shall say that $T$ admits a generalized Kato-meromorphic decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and the reduction $T_N$ is meromorphic. In this paper we shall investigate such kind of operators and corresponding spectra, the generalized Drazin-meromorphic spectrum and the generalize…
We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fréchet-Urysohn in weak* topology. We also introduce another concept of perturbation called u…
In \cite{koliha}, Koliha proved that $T\in L(X)$ ($X$ is a complex Banach space) is generalized Drazin invertible operator equivalent to there exists an operator $S$ commuting with $T$ such that $STS = S$ and $σ(T^{2}S - T)\subset\{0\}$ which is equivalent to say that $0\not\in \mbox{acc}\,σ(T).$ Later, in \cite{rwassa,rwassa1} the authors extended the class of generalized Drazin invertible operators and they also extended the class of pseudo-Fredholm operators introduced by Mbekhta \cite{mbekhta} and other classes of semi-Fredholm operators. As a continuation of these works, we introduce and …
Motivated by the questions in the theory of Fredholm stability in Banach space and Kato's strictly singular operators we answer several natural questions concerning ``orthogonality'' in normed spaces and the properties of metric projections. What the reader will see below might have benn known long ago, but we did not find it in the literature. Some open (for us) questions are formulated at the end of Sections 7 and 9.
We prove that for any Bernstein function $ψ$ the operator $-ψ(A)$ generates a holomorphic $C_0$-semigroup $(e^{-tψ(A)})_{t \ge 0}$ on a Banach space, whenever $-A$ does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that $(e^{-tψ(A)})_{t \ge 0}$ is holomorphic in the holomorphy sector of $(e^{-tA})_{t \ge 0},$ and if $(e^{-tA})_{t \ge 0}$ is sectorially bounded in this sector then $(e^{-tψ(A)})_{t \ge 0}$ has the same property. We also obtain new sufficient conditions on $ψ$ in order that…
An interesting result by T. Kato and A. Pazy says that a contractive semigroup (T(t)) on a uniformly convex space X is holomorphic iff limsup_{t \downarrow 0} ||T(t)-Id|| < 2. We study extensions of this result which are valid on arbitrary Banach spaces for semigroups which are not necessarily contractive. This allows us to prove a general extrapolation result for holomorphy of semigroups on interpolation spaces of exponent θ in (0,1). As application we characterize boundedness of the generator of a cosine family on a UMD-space by a zero-two law. Moreover, our methods can be applied to R-secto…
This is a survey paper concerned with strongly continuous semigroups in a Banach algebra (often itself simply the algebra of bounded linear operators on a Banach space). These are defined either on $(0,\infty)$ or on a sector in the complex plane, in which case they are supposed analytic. However, the semigroups are not assumed to be strongly continuous at 0. The results in this survey, beginning with classical theorems of Beurling, Kato and Neuberger and concluding with recent work by the authors, are of the nature of "0-1" laws, indicating that the quantitative behaviour of the semigroup at …
Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows: Let $ϕ:S^{2} \rightarrow S^{2}$ be a homeomorphism of order n and $λ\neq 1$ be an nth root of the unity, then for every complex valued continuous function $f$ on $S^{2}$ the function $\sum_{i=0}^{n-1} λ^{i}f(ϕ^{i}(x))$ must be vanished at some point of $S^{2}$. We give a generalization in term of action of compact groups. We also discuss about some noncommutative versions of the Borsuk- Ulam theorem
In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space.
For every well founded tree $\mathcal{T}$ having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a $\mathcal{L}_\infty$-space $X_{\mathcal{T}}$. We prove that for each such tree $\mathcal{T}$, the Calkin algebra of $X_{\mathcal{T}}$ is homomorphic to $C(\mathcal{T})$, the algebra of continuous functions defined on $\mathcal{T}$, equipped with the usual topology. We use this fact to conclude that for every countable compact metric space $K$ there exists a $\mathcal{L}_\infty$-space whose Calkin algebra is isomorphic, as a Ban…
L.Accardi shows that the Banach space of singed measures is homeomorphic to the Hilbert space formed by so-called root measures. In this paper, we redefine root measures in view of the theory of measures on infinite dimensional spaces, and introduce a notion of differentiation, Fourier transform, and convolution product for root measures, and examine those relations. We also study about Lévy Laplacian on Wiener space as application. It is shown that the symbol of Lévy Laplacian is equal to the quadratic variation of paths.
An asymptotic stability result for parabolic semilinear problems in $L_2(Ω)$ and interpolation spaces is shown. Some known results about stability in $W^{1,2}(Ω)$ are improved for semilinear parabolic mixed boundary value problems. The approach is based on Amann's power extrapolation scales. In a Hilbert space setting, a better understanding of this approach is provided for operators satisfying Kato's square root problem; as a side result some equivalent characterizations of these operators are obtained.
Due to the seminal works of Hochbruck and Ostermann exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than $π/2$ (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing on the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the sem…
Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in $L^2$ spaces, and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of $L^p$ spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator, to prov…
Let H be a real (or complex) Hilbert space. Every nonnegative operator $L \in L(H)$ admits a unique nonnegative square root $R \in L(H)$, i.e., a nonnegative operator $R \in L(H)$ such that $R^{2}= L$. Let $GL^{+}_{S}(H)$ be the set of nonnegative isomorphisms in $L(H)$. First we will show that $GL^{+}_{S}(H)$ is a convex (real) Banach manifold. Denoting by $L^{1/2}$ the nonnegative square root of $L$. In [10], Richard Bouldin proves that $L^{1/2}$ depends continuously on $L$ (this proof is non-trivial). This result has several applications. For example, it is used to find the polar decomposit…
Let L_U = -Delta+U be a Schrödinger operator on R^d, where U\in L^1_{loc}(R^d) is a non-negative potential and d\geq 3. The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\in L^1(R^d): \|f\|_{H^1(L_U)}:= \|sup_{t>0} |K_{t,U} f| \|_{L^1(R^d)} < \infty. Assume that U=V+W, where V\geq 0 satisfies the global Kato condition sup_{x\in R^d} \int_{R^d} V(y)|x-y|^{2-d} < \infty. We prove that, under certain assumptions on W\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. An atom a for H^1(L_U) is eith…
In this paper, we study the differentiation operator acting on discrete function spaces; that is spaces of functions defined on an infinite rooted tree. We discuss, through its connection with composition operators, the boundedness and compactness of this operator. In addition, we discuss the operator norm and spectrum, and consider when such an operator can be an isometry. We then apply these results to the operator acting on the discrete Lipschitz space and weighted Banach spaces, as well as the Hardy spaces defined on homogeneous trees.
In a Banach space $X$ the linear difference equation with constant coefficients $x_{n+p} = a_1x_{n+p-1} +\ldots + a_px_n,$ is Ulam stable if and only if the roots $r_k,$ $1\leq k\leq p,$ of its characteristic equation do not belong to the unit circle. If $|r_k| > 1,$ $1\leq k\leq p,$ we prove that the best Ulam constant of this equation is $ \frac{1}{|V|}\sum\limits_{s=1}^{\infty}|\frac{V_1}{r_1^s}-\frac{V_2}{r_2^s}+\ldots +\frac{(-1)^{p+1}V_p}{r_p^s}|,$ where $ V = V (r_1, r_2, \ldots, r_p)$ and $V_k =V (r_1,\ldots, r_{k-1}, r_{k+1},\ldots, r_p),$ $1\leq k\leq p,$ are Vadermonde determinants.
Nonlocal problems for Laplace equation in Bochner spaces
Yildiz Technical University（伊尔迪兹理工大学）；Institute of Mathematics, The Ministry of Science and Education（阿塞拜疆教育部数学研究所）；Azerbaijan University of Architecture and Construction（阿塞拜疆建筑与建设大学）；Odlar Yurdu University（奥德拉尤尔杜大学）；University of Salerno（萨莱诺大学）
For a Banach space $B$ and for a class $\A$ of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements $A \in \A$ can be chosen to depend continuously on $A$, whenever nonconvexity of each $A \in \A$ is less than $\f{1}{2}$. The key geometric argument is that the set of all uniform retractions onto an $\a-$paraconvex set (in the spirit of E. Michael) is $\frac{\a}{1-\a}-$paraconvex subset in the space of continuous mappings of $B$ into itself. For a Hilbert space $H$ the estimate $\frac{\a}{1-\a}$ can be improved to $\frac{\a (1+\a^{2})}{1-\a^{2}}$…
